Lectures on Elementary Probability

6.5. EXPONENTIAL RANDOM VARIABLES 41
Such an integral is called a Γ(m + 1) integral. The result follows from the
identity and the obvious fact that Γ(1) = 1.
It is worth recording that there is also an explicit formula for the Gamma
function for half-integer values.
Theorem 6.5 The value of Γ(m + 1 2
) for natural numbers m = 0, 1, 2, 3, . . . is
given by
Γ(m + 1 2 ) = (m − 1 2 )(m − 3 2 ) · · · 3 1√ π. (6.30)
2 2
Such an integral is called a Γ(m + 1 2 ) integral. The result follows from the
identity and the non-obvious fact that Γ( 1 2 ) = √ π. The reason for this last fact
is that
Γ( 1 ∫ ∞
∫ ∞
2 ) = t − 1 2 e −t dt = 2 e −w2 dw = √ π. (6.31)
0
The last equality comes from the normalization integral for a Gaussian (normal)
density with variance parameter 1/2.
6.5 Exponential random variables
Consider a Poisson process. Let W be the waiting time until the first jump.
This kind of random variable is called exponential with parameter λ. Then since
the event W ≤ t is the same as the event N(t) ≥ 1, we have the following result.
Theorem 6.6 Let W be an exponential random variable with parameter λ.
Then
P [W ≤ t] = 1 − e −λt (6.32)
for t ≥ 0.
The following theorem may be proved by differentiating. However we give
another more direct proof below.
Theorem 6.7 For an exponential random variable W with parameter λ the
probability density is given by
for t ≥ 0.
0
f(t) = λe −λt (6.33)
Proof: If the waiting time for the first jump is in the interval from t to
λdt, then there were no jumps up to time t, and then there was a jump in the
following tiny interval. Thus
P [t < W < t + dt] = P [N(t) = 0]λ dt. (6.34)
Theorem 6.8 The expectation of an exponential random variable with parameter
λ is
E[W ] = 1 λ . (6.35)